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Mirrors > Home > MPE Home > Th. List > om2noseqlt2 | Structured version Visualization version GIF version |
Description: The mapping 𝐺 preserves order. (Contributed by Scott Fenton, 18-Apr-2025.) |
Ref | Expression |
---|---|
om2noseq.1 | ⊢ (𝜑 → 𝐶 ∈ No ) |
om2noseq.2 | ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)) |
om2noseq.3 | ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω)) |
Ref | Expression |
---|---|
om2noseqlt2 | ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 ∈ 𝐵 ↔ (𝐺‘𝐴) <s (𝐺‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | om2noseq.1 | . . 3 ⊢ (𝜑 → 𝐶 ∈ No ) | |
2 | om2noseq.2 | . . 3 ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω)) | |
3 | om2noseq.3 | . . 3 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω)) | |
4 | 1, 2, 3 | om2noseqlt 28320 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 ∈ 𝐵 → (𝐺‘𝐴) <s (𝐺‘𝐵))) |
5 | 1, 2, 3 | om2noseqlt 28320 | . . . . 5 ⊢ ((𝜑 ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ ω)) → (𝐵 ∈ 𝐴 → (𝐺‘𝐵) <s (𝐺‘𝐴))) |
6 | 5 | ancom2s 650 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 ∈ 𝐴 → (𝐺‘𝐵) <s (𝐺‘𝐴))) |
7 | fveq2 6907 | . . . . 5 ⊢ (𝐵 = 𝐴 → (𝐺‘𝐵) = (𝐺‘𝐴)) | |
8 | 7 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 = 𝐴 → (𝐺‘𝐵) = (𝐺‘𝐴))) |
9 | 6, 8 | orim12d 966 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) → ((𝐺‘𝐵) <s (𝐺‘𝐴) ∨ (𝐺‘𝐵) = (𝐺‘𝐴)))) |
10 | nnon 7893 | . . . . 5 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On) | |
11 | nnon 7893 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
12 | onsseleq 6427 | . . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ⊆ 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))) | |
13 | ontri1 6420 | . . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ 𝐵)) | |
14 | 12, 13 | bitr3d 281 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) ↔ ¬ 𝐴 ∈ 𝐵)) |
15 | 10, 11, 14 | syl2anr 597 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) ↔ ¬ 𝐴 ∈ 𝐵)) |
16 | 15 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) ↔ ¬ 𝐴 ∈ 𝐵)) |
17 | 1, 2, 3 | om2noseqfo 28319 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:ω–onto→𝑍) |
18 | fof 6821 | . . . . . . . 8 ⊢ (𝐺:ω–onto→𝑍 → 𝐺:ω⟶𝑍) | |
19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐺:ω⟶𝑍) |
20 | 3, 1 | noseqssno 28315 | . . . . . . 7 ⊢ (𝜑 → 𝑍 ⊆ No ) |
21 | 19, 20 | fssd 6754 | . . . . . 6 ⊢ (𝜑 → 𝐺:ω⟶ No ) |
22 | 21 | ffvelcdmda 7104 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ ω) → (𝐺‘𝐵) ∈ No ) |
23 | 22 | adantrl 716 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐺‘𝐵) ∈ No ) |
24 | 21 | ffvelcdmda 7104 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐺‘𝐴) ∈ No ) |
25 | 24 | adantrr 717 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐺‘𝐴) ∈ No ) |
26 | sleloe 27814 | . . . . 5 ⊢ (((𝐺‘𝐵) ∈ No ∧ (𝐺‘𝐴) ∈ No ) → ((𝐺‘𝐵) ≤s (𝐺‘𝐴) ↔ ((𝐺‘𝐵) <s (𝐺‘𝐴) ∨ (𝐺‘𝐵) = (𝐺‘𝐴)))) | |
27 | slenlt 27812 | . . . . 5 ⊢ (((𝐺‘𝐵) ∈ No ∧ (𝐺‘𝐴) ∈ No ) → ((𝐺‘𝐵) ≤s (𝐺‘𝐴) ↔ ¬ (𝐺‘𝐴) <s (𝐺‘𝐵))) | |
28 | 26, 27 | bitr3d 281 | . . . 4 ⊢ (((𝐺‘𝐵) ∈ No ∧ (𝐺‘𝐴) ∈ No ) → (((𝐺‘𝐵) <s (𝐺‘𝐴) ∨ (𝐺‘𝐵) = (𝐺‘𝐴)) ↔ ¬ (𝐺‘𝐴) <s (𝐺‘𝐵))) |
29 | 23, 25, 28 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (((𝐺‘𝐵) <s (𝐺‘𝐴) ∨ (𝐺‘𝐵) = (𝐺‘𝐴)) ↔ ¬ (𝐺‘𝐴) <s (𝐺‘𝐵))) |
30 | 9, 16, 29 | 3imtr3d 293 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (¬ 𝐴 ∈ 𝐵 → ¬ (𝐺‘𝐴) <s (𝐺‘𝐵))) |
31 | 4, 30 | impcon4bid 227 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 ∈ 𝐵 ↔ (𝐺‘𝐴) <s (𝐺‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 ↾ cres 5691 “ cima 5692 Oncon0 6386 ⟶wf 6559 –onto→wfo 6561 ‘cfv 6563 (class class class)co 7431 ωcom 7887 reccrdg 8448 No csur 27699 <s cslt 27700 ≤s csle 27804 1s c1s 27883 +s cadds 28007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-nadd 8703 df-no 27702 df-slt 27703 df-bday 27704 df-sle 27805 df-sslt 27841 df-scut 27843 df-0s 27884 df-1s 27885 df-made 27901 df-old 27902 df-left 27904 df-right 27905 df-norec2 27997 df-adds 28008 |
This theorem is referenced by: om2noseqiso 28323 |
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